Interpolation Letween weighted U-spaces

Interpolation Letween weighted U-spaces

Jo~N E. GILBERT University of Texas

1 . I n t r o d u c t i o n

One of the main problems in a n y application of interpolation space theory is the identification of the interpolation space (~o, ~ 1 ) ~ generated from a pair of Banach spaces c2(~0, ~ 1 by a given interpolation m e t h o d ~. The importance of interpolation space theory stems in part from the frequency with which (~o, fiF1)r is identified with classical Banach spaces and the deeper understanding the theory t h e n brings to these spaces. Applications to Lorentz Leq-spaces a n d to Lipschitz spaces are g o o d e x a m p l e s (cf. [4], for example).

In this paper the interpolation spaces (~0,/~o~,)o, ~ generated b e t w e e n weighted LP-spaces b y the (real) J-, K - m e t h o d s of Peetre will be characterized. W i t h o u ~ real loss of generality w e shall a s s u m e ~o o ~ 1. T h e ~)diagonab) spaces (L~0,/~o~1)o,v, it is k n o w n already, coincide with another weighted LV-space:

( L v v 1 - o o 0 < 0 < 1

. . . . , Lo>,)o,~ = L ~ , ~' = ~ o ~o~,

([7], [15]) a n d the associated interpolation t h e o r e m reduces to the Stein-Weiss extension of the classical IV[. Riesz theorem to spaces with changes of measure.

Peetre began the characterization of the ~)off-diagonab) cases by identifying v 1 < p < oo, with one of a family of spaces introduced by Beurling (L~o, L~)o, ~,

([3]) in connection with problems of spectral synthesis. I-Ierz later generalized Beurling's definition and considerably clarified Beurling's paper though without systematic recourse to interpolation space theory ([11]). I n sections 4 and 5 we complete the characterizations of (LV~o, LV)o,q, 1 _~ q _~ oo, and identifications with the Beurling spaces. With these characterizations as well as the theory of (homogeneous) Besov spaces the various results of ~ e r z can be obtained very easily using simple interpolation space techniques (see [6]).

The Beurling spaces have been considered in m a n y contexts other t h a n specbral synthesis (cf. [9], [10], [12], [13], [18]) with various characterizations being obtained

2 3 6 J O H N E. G I L B E R T

or used. I n section 3 we obtain one simple characterization (theorem (3.1)) from which all characterizations follow as special eases. This theorem, in addition, com- pletes other results of Peetre ([17] theorem (2.1)). Since the only property of LP- spaces needed is the fact that Le is an Leo-module via pointwise multiplication, the results in w 3 are presented from the point of view of Banach Function spaces

~ ; however, this section can be read equally well substituting Le for ~)d.

Some applications of these results to harmonic analysis appear in [1] and [2].

2. Elements of interpolation space theory

Wherever possible the current notation and terminology of abstract inter- polation space theory will be used (cf. [4], [7], [14]). Let ~)r ?)dl be Banach spaces both continuously embedded in some t t a u s d o r f f topological vector space so that then (9(7o, ~)~1) forms a comloatible couple in the sense of [7] chap. 1. Function norms (K(t, (.)) and J(t, (.)), 0 < t < oo, are defined on the sum of ~)r + ?/F1 and intersection ~ 0 g l c9r 1 b y

K(t, f ) -~ K(t, f; ~So, ?~1) = inf (ilf011x, + t]lf~li~r,),

S = f ~ + f l

J(t, g) = J(t, g; ~So, ~ = m ~ x (IIgli,:r0,

ttlgll~.,)

where f e ~So + o_r~ and g e ~ o Cl ~m~ respectively. These function norms are both continuous and monotonic increasing in t ([4] p. 167). The interpolation space (~m0, mr is the subsp~oe of ~ o + ~r~ of all f for which

co

lif[to, q~K : ( t - ~ q 0 = O, 1, q ~- oo, (1)

0

is finite (obvious modifications when q ~ oo); under the norm I](.)][o.q;K(c)~o, 9~)o,~; K is a Banach space, non-trivial whenever 0, q are restricted as in (1). The interpolation space (~o, ~ ) o , ~ ; j consists of all f in 9(7 o + ~3(~ for which there is a strongly measurable function u: (0, oo) --> ~ o 17 ~(~1 satisfying

co co

f f

(i) K(1, u(t)) ~ < ~ , (ii) f -= u(t) 7 ' (2)

0 0

(iii) (t-~ u(t)))~ ~ ~ ' 0 • 0, 1, q = 1. (3)

0

I f we set

I 2 { T E I % P O L A T I O I ~ ] 3 E T ~ ' E E N ~ T E I G H T E D L P - S ? P A C E S 237

co

Ilfllo,q;j

0

the infimum being taken over all u satisfying (2) and (3), then (~o, ~(~l)o,~;J becomes a non-trivial Banach space. I t is known t h a t

( ~ 0 , ~1)o,++ = (~)~0, ~ ) o , ~ ; K , 0 < 0 < 1, 1 < q < oo , at least up to equivalence of norms.

These spaces (~(~0, ~)o.~;K, (~(?0, ~)(~l)o,q;j can b e constructed in exactly the same w a y to yield the same spaces (with equivalent norms) b y replacing the multi- plicative group /~. : (0, oo) with the multiplicative discrete group {r~: n E Z}, r being fixed, r > 1, the H a a r measure dt/t on (0, ~ ) being replaced b y Haar measure on {r~: n E Z}. We shall refer to this m e t h o d as the discrete method of construction (see [6] or [14] for full details). The Lp-spaces on /~. defined with respect to H a a r measure will be denoted by L~.

The previous methods of construction are all >>real>> methods in contrast to the

>>complex>> m e t h o d introduced b y Calder6u et al. (cf., for instance, [5]). I n this method, to each compatible couple (-~0, ~ 1 ) is associated a Banach space [~o, 9(~1]o, 0 < 0 < 1. Since we are concerned exclusively with characterizations of inter- polation spaces obtained b y the J-, K-methods further details are omitted. Let us recall only t h a t if (X, #) is a totally a-finite measure space and Lp ( = Lv(X,/Z)) the usual Lebesgue spaces, t h e n

(L p~ L )o.q;K ~ LP~, [L v~ P~ ^{Lpl]o : L p}

where 1/p 9 : (1 -- 0)/290 q- 0/291 and Lr~(~ LPq(X, #)) is the usual Lorentz space associated with (X,/z) (see [4] w 3.3.1, [5]).

3. Interpolation between weighted function spaces

For a totally a-finite measure space (X,/z), 9~/(X, #) denotes the space of /z-measurable complex-valued functions on X. A ]3anach space 9g of (equivalence classes of) functions in ~/~ is called a Banach function s29ace provided

(a) if" gECl/]i(X,/z) and tg] ~ If] /z-a.e. for some fEP(~ then g belongs to a n d llgll:~ -< Jl/ll~,

(b) if {f,} is a sequence of non-negative functions in ~(~ and fn ~f/z-a.e. then sup llfoll.~ = Ilfll~.

n

Obviously the Lebesgue LP-spaces:

[Iflrp = If(x)lpd/z , 1 __<29 ~ ~ , llf[ico = ess. sup. If(x)l (4)

X

238 f f O H ~ E . G I L B E R T

are Banach Function spaces as are the Lorentz LPq(X,/x) spaces, condition (b) being the classical F a t o u property. Notice that (a) ensures ~ is an L ~ ( X , g)- module via pointwise multiplication and

Ilfg[l~ < Iifll~ Ilgll~, f ~ ~ , g e L | .

When co is a function in c~f/(X,/~) with co > 0/x-a.e., ~(~ denotes the weighted function space of all (equivalence classes of) functions f in c~(X,/1,) for which fco E 9(~; we set

For weighted Z~(X, co) spaces we shall write

[l/llp, o = [[/collp, 1 < p ~ ~ ,

with

II(.)ll~

where

m ( x ) ~-- rain (1, co(x)), M ( x ) -= max (1, co(x))

(cf. [15]). In this and the succeeding section we shall characterize the interpolation spaces

( ~ , ~.)o,q~K, (L p, L[)o,q;K, For the complex method the interpolation space

0 < 0 < 1 , l < q _ < ~ .

[ Le,/~o,]o is known:

[jLp, Lo,]o = L~, v(x) : co(x) ~ , P

(see [5] pp. 123--4 for even more general results). As frequently happens, however, the real methods yield directly a richer family of spaces.

Throughout, a will denote a non-negative, piecewise continuous function in L~(0, ~ ) such t h a t ta(t) belongs to L~(0, ~); thus, with o, defined for each fixed t b y

at().) = t2a(t)O, t, 2 C (0, ~ ) , our assumptions on a ensure that

max (llo[]~, sup ]la,[[~) < ~ . (5)

$

Furthermore, the composition at o co belongs to L~(X, #) and f . (a, o co) to 9C whenever f C ~ . All of our characterizations follow (more or less easily) from the following result.

I N T E R P O L A T I 0 1 ~ B E T W E E N W E I G H T E D L P - S P A C E S 239 THEOREM (3.1). There exist constants depending only on (~, 0 and q such that

0

for all

f

dt ) ~/~

o w)ll~:)q - [ < const. Ilfllo,.,K

Proof. W h e n f = f0 + f l is a decomposition of f in ~ + ~)C,

It f " (~, ~ co)lla~ _< lifo" (o, o ~o)lla~ -t- ]IA" (~, o ~o)ll~ c _< (sup II~,ll~)IIfolla~ + t(ll~lloo)llficoll,r,

#

a n d so, in v i e w o f (5),

Ill" (~ o co)II~ c ~ m a x (sup I[~,[Io~, II(~lE)K(t,f) (6)

$

for all t E (0, oo). The right hand i n e q u a l i t y follows.

N o w let V be a non-negative c o n t i n u o u s f u n c t i o n with c o m p a c t s u p p o r t in (0, oo). T h e n the f u n c t i o n u = u(t):

u(t) = f ( x ) . a,(co(x)) 9 ~(to~(x)), t r (0, ~ ) , x e X is a m e a s u r a b l e 9(~ ~l 9(~o~-valued function such t h a t

Ilu(t)ll~ ~ II~ll~llf ~ (~, o co)][~

while

Thus u:

tHu(t)~~ = ]l f" (a, o w) . (~, o w)ll~ _< (sup [T~,ll~)llf" (a, o s r .

t

(0, oo) ---> ~_~ N ~)~ and

J(t, u(t)) ~__ m a x ([l~T[r sup Ilk, it)Ill" ((~, ~ o))II~ 9

$

B u t with a suitable normalization,

co

f

I = u(t) 7 "

0

The left h a n d i n e q u a l i t y o f the t h e o r e m n o w follows.

I f the discrete construction is u s e d in the p r o o f of t h e o r e m (3.1) the following

~)diserete~) characterization of (~(~, 9(~)o.q;K is obtained.

TH~.O~EM (3.2). For each fixed r > 1 there exist constants depending only on O, q and (~ such that

240 J O H N E , G I L B E I ~ T

const.

Ilfllo,,;J ~ ( ~

(a,~o co)lice)q) 1/q

- - c o

for all

f

The first application of theorems (3.1) and (3.2) gives what in practice is the most useful characterization of (cS, 9(~)o,q; ~. For fixed r > 1 denote by 9 the function

l

T(t) = - / , 1 < t < r O, elsewhere.

(7)

When X , - - { x E X : o ) ( x ) < 1/t} and Z, is the characteristic function of X,, then Ttoo4= ~t/r-- Zt and

A(z~k) = z,~ o o = Z,~-~ - - Z, k.

T~EOBEM (3.3). For each r > 1 there exist constants depending only on 0 and q, 0 < 0 < 1, 1 < q < oo, such that

for all

f

eonst.

Ilfllo,~;j ~ ( ~

const. Ilftlo,,,K

( ~ , ~)e,o)o,~;K.

Proof. The particular choice (7) of ~ satisfies (5) and theorem (3.2) then applies.

Now define "r, "d by

1, O < t < l , { 1

T ( t ) = ~ ' ( t ) = ~-, t ~ l ,

O, elsewhere, O, elsewhere.

Obviously ~ and ~' satisfy (5)while

?

" r 1 7 6 1 6 2 1 7 6 ~ t e ~ z t , "c, o o 9 ~ 1 - - Z t ,

and

tI]fo~z,[Ioc ~ K ( t , f ) , llf"

(1 -- z,)ll~ ~ K(t,f)

(el. (6)). Theorem (3.1) thus gives:

(8)

T~EOBEM (3.4). When Z, is the characteristic f u n c t i on of the set X t = {x C X:

o)(x) ~ l/t} then

I N T E I % F O L A T I O N B E T W E E N AVEIGI-ITED L P - S r A O E S 241

03 03

t } '

0 0

define equivalent n o r m s on ( ~ , ~ 0 < 0 < 1, 1 < q < co.

The Lipschitz character of ( ~ , ~o,)o,q;K can be described b y setting

~(t) = e-', r = T (e-' - 1), 1 ⁽⁹⁾

so t h a t 7, z' s a t i s f y (8) a n d

t e _ t ~ o

T t o ~ : t e o e - t ~ ~:t ^{o o ~ :} - - 1 .

THEOREM (3.5). F o r 0 < 0 < 1 a n d 1 < q < oo,

03 cO

(f(,o,,z

0 0

define equivalent n o r m s on (5)C, ~(~)o,~;K.

1)II,~)q

Proof. A p p l y t h e o r e m (3.1) w i t h 7, z' d e f i n e d b y (9).

I t is of i n t e r e s t to f o r m u l a t e t h e characterizations c o n t a i n e d in t h e o r e m s (3.4) a n d (3.5) for t h e special case r = L P ( X ) , 1 < 29 < oo. W h e n a = (l/t) rain (1, t) in t h e o r e m (3.1) a n d 1 < T < 0%

03 co

f

f f

0 0 X

f lf(x)t'o(x)~ t-'~

t)'{ t)

X 0

I n ease p = co t h e analogpus result holds:

sup t-~ 9 (~, o co)!Io 3 = I[fio*l103 . sup (t - ~ rain (I, t ) ) .

t $

Consequently, for all p , l ~ 20 ~ o%

03

(f

t)llLp,

0

which establishes (cf. [7] p. 25, 31; [15]):

2 4 2 J O H N ]7]. G I L B E R T

THEOREM (3.6). l~or 0 < 0 < 1 and 1 ~ p ~ co

(L'(X), LUX))o,~,~ = {f: ( f lf(x)l'o(x)~ v"

x

and

For non-diagonal values of q,

const,

llfllo.p,.r ~ Ilf~~ ~

Ilfllo,,;~.

i.e., q ve ~o, theorems (3.3) and (3.4) yield:

THEOrEm[ (3.7). The expressions

(i, If(x)]Pd/-*J ) ) , r > l ,

< ~o(x)_< t k

(// "):;

(ii) t'-~ f

to,(x) _< 1

(i(Iz

:)"

(iii) t -~

If(x)l"d,. ] )

to~(x)>l

(LP(X), L~(X))o,~: K for all 0 < 0 < 1, define equivalent norms on

and 1 <_p ~ oo.

l < q < oo,

Remark. Partial results along the lines of theorem (3.7) were obtained b y Peetre in [17] (of. p. 63).

4. Beurling spaces

We come now to our main characterization of the interpolation spaces (/F(X), JLP~(X))o.~:K, identifying these spaces with important Banaeh spaces introduced by Beurling ([3]).

With 7 ---- ( l i p -- 1/q) and 0 < 0 < 1 denote b y tg(0, 7) the class of functions

~0 in L 1, ~ 0 > 0 , such that

(a) ]I~NL~ = 1, (b) t~ ~ is monotonic increasing.

Such functions ~ certainly exist; for instance, suitable normalizations of 9(O = t-~ m i n (1, tlh), 7 r 0, q~(t) = t -~ min (1, t), y - = 0 satisfy (a), (b). For fixed (o we t h e n define B~o,q(X, co) by

I N T E R P O L A T I O N B E T W E E N W E I G H T E D Z P - S P A C E S 2 4 3

B g . J X , ~) = t J {L:(X): w = ~~ o ~),}. r -< 0 , r

B ~ , ~ ( X , ~ ) = f l { Z ~ ( X ) : w = ~ ~ r >--0"

Cea(o,v)

I n [3] additional convolution properties are imposed on ~ to ensure t h a t certain of the B~o,q(X, co) spaces are Banach algebras. Discussion of this particular facet will be omitted from this paper.

THEOI~EM (4.1). Let ~ = l i p - - 1/q where 1 ~_ 19 < oo and 1 ~_ q ~ o~, T h e n for each O, 0 < 0 < 1, there exist constants such that for all f i n ( I ~ ( X ) , L~(X))o.~; K

eonst. [Ifllo.q~j ~ i n f llfo~~ o o~)~ll~ ~ const. Ilfllo;q;K

~eo(o,r) when ~ ~ 0, and

when ~ ~ O.

(10)

const. ]lfllo,~j ~ sup IIf~~ o co)'llp ~ const. I[fL, q.,K (11)

~eo(o,r) I n 19articular,

(L~(X), ~(X))o.~;~ = Bg, q(X, co).

Proof. (i) We establish first the left hand inequality in (10). Since 7 ~ 0, automatically q ~ p < oo.

Choose a n y non-negative continuous function ~ with compact support in [1, oo), 7J ~ 0, and define a on (0, oo) b y a ( t ) = (1/t)~p(t). I n view of theorem (3.1) the left h a n d inequality in (10) follows provided

( t - ~ (~r, o o~)[Ip)q _< const, i n f ]If" c~176 ~ ~)~IIP 9 (12)

0 9

But, since t~ r is monotonic increasing, t-~ o co)(x) = t-~ < co(x)~

because ~o(t~o(x)) is non-zero only when to~(x) > 1 and t h e n ~(to~(x)) 3> O. T h u s t - ~ 9 (~, o co)liP _< T(1/t)-rl[fw~ o ~)'r(t~)ll~

and so

f(f )(1),

f

0 0 X

Using crucially the fact that q < 19, we deduce from 1-151der's inequality applied to (13) that the left hand side of (12) is dominated by

2 4 4 a o ~ ~ . G I L B E R T

c o

(f(f

) ^~)l/p

0 x

inverting t h e order of i n t e g r a t i o n in (14). This establishes (12).

(ii) I n proving t h e righ~ h a n d i n e q u a l i t y in (10) we m a y assume q < p ( < ~ ) , i.e., y < 0, since t h e case q = p, i.e., y = 0, has been t a k e n care o f a l r e a d y in t h e o r e m (3.6). F o r f i x e d f in (L p,/~,o)o,~;~c define ,p on (0, ~ ) b y

l#(t)r = t-o 2-OqK(2, f ) , - e .

1/t

K ( 2 , f ) <_ m a x (1, 2 t ) K ( 1 / t , f ) , Since

1/t

I-Ience,

+ oo= Ito.(:

F o r this f u n c t i o n ~,

x 11o4~)

(p+ Oq--q) -1~.

f If ]~ f ~

= 2-O~K(2,f)~ -~ If[p(1 -- z~)d# ~ - ~ 2-+qK(2,f) ~ -~- = (Hf[Eo,~;K) ~

x o

(of. (8)). N o w set ~v = ~v/II~oIIL~. T h e n 9 belongs to Q(0, y) a n d ][fco~ o ~oYIb _< (IIflIo,+K)VP(IIWIIL,) -~ <-- eonst. [ifllo,+K completing t h e p r o o f of (10).

(iii) W e t u r n n e x t to t h e left h a n d i n e q u a l i t y in (11). F i r s t we assume 2o < q < co.

I t is t h e n enough to show t h a t

c o

(/

o + e t~(o, ~,)

N o w the left h a n d side of (15) is

c o

s2p(f g(,)t-~ tfL~(l-z,)~,)~-')

0 X

I N T E R P O L A T I O N B E T W E E N W E I G H T E D ~ P - S I ~ A C E S 245 t h e s u p r e m u m being t a k e n over all g ~ 0

(remember P7 -+- 1~(q/p) -- 1.

where

such t h a t

co

f g(t)~/p ~ d t _ 1

t

0

B u t , w i t h a change in t h e order of integration,

co

f g(t)t-~ ( f lf[~O - x,)dff ) y =

0 X

f lflP~o~ o co)'Pdff

X

Clearly t~ ~ is m o n o t o n i c increasing. Also, b y (16),

co co co co

_

t)l:p'dq'~<(f g(t)i/,,,~"'(f a-o~,

0 1 0 1

regarding f ~ g(2/t)2-~ as a convolution on _R.. Thus w i t h ~ = t0/]lt01IL 1 ,

/

(/

(:

)

X

< const. [ sup (llf~~ o ~ , ) % ) q e ~(o, r)

for all g satisfying (16). This establishes (15).

I n case q = co we m u s t replace (15) b y

s u p ^{( t - ~ 9} (1 - - X,)l[.) --~ c o n s t , s u p tlfio~ o ~o)~llp 9 ( 1 7 } T h e p r o o f o f (17) is e n t i r e l y analogous t o t h a t o f (8) if we r e m e m b e r t h a t 297 = 1 w h e n q = oo.

(iv) F i n a l l y we establish t h e r i g h t h a n d side o f (11). Choose a n y n o n - n e g a t i v e continuous f u n c t i o n o, ~ ~ 0, w i t h s u p p o r t in (0, 1]. Then, because t~ ~ is m o n o t o n i c increasing,

co

~oP~ o co) p' < const. (~, o (o)Pt-P~ -~,

0

t h e c o n s t a n t being i n d e p e n d e n t o f ~. I n this ease, w i t h a change in t h e order o f integration,

y~(t)r = t -~

( ] 6 )

co a0

1/t 1

246 ffOHN ^{E .} GILBERT

co

f

f (f

)

X 0 X

(/

< eonst. (II~II~)P' (t-~ 9 (a, o r

0

b y tISlder's inequality using crucially the fact t h a t q ~ p. This establishes the right hand side of (11) completing the proof of theorem (4.1).

On Lv(X), 1 ~ la < oo, the semi-group {M(t): 0 • t < oo} of multiplication operators:

M(t): f(x) ---> e-'~ f e LP(X),

is a semi-group of contraction operators of class (q~0) whose infinitesimal generator Ao~ is the multiplication operator A~o: f--> -- wf (cf. [4] p. 3 for definitions). The domain of definition D(A~o ) of A~ becomes a Banaeh space under the graph norm

![fllD(A)o , = IlfJlp + [l~fllp, f c L P ( X ) .

Thus, if co(x) ~ ~ > 0 on X as frequently is the ease in applications, L~(X) is norm equivalent to D(Ao). The general t h e o r y of the c o m m u t a t i v i t y of inter- polation funetors as developed b y Grisvard ([8] pp. 169, 171) enables us to identify the respective complex and real interpolation spaces

[B~:q~ ~o), BPol.q,(X, ~o)]~, (B~:q(X, r B~2~(X, o~))~.q;~:

with

1 ~Po, P l < oo, 0 ~ 0 0 , 0 1 , 0 , ~ 1, 1 ~qo, ql, q ~ oo. (18) THEOR]~ (4.2). Under the restrictions (18),

B v" ~ X o.,So, , o~), Bo..~,(X, ~ ) L P' = B ~ q ( X , o~)

and

where in both cases

1~ = B g J X , o~)

,tB v'~,~,cX , ~o), B~,~(X, o~))~,~; K ,

0 = (1 - ~)0 o + # ~ .

Remark (4.3). I f 09 is not bounded a w a y from 0 on X the (non-homogeneous) Lipschitz spaces used by Grisvard have to be replaced with Lipschitz spaces of

I : N T E R P O L A T I O ~ ~ B E T W E E N W E I G H T E D LP-SPACES 247 homogeneous character (cf. [16] p. 286). Though important in applications we shall omit a n y such considerations here; the theory of such spaces is developed syste- matically in [6], for instance, to which the reader is referred.

5. Examples

We shall consider briefly the case when X =/~n, /~ is Lebesgue measure on R" and o~(x)~-- ( x ~ - ~ . . . - ~ - x ~ ) " / 2 ~ lxt~; the case of X = Z n, o~(m)= Iml ~ is entirely analogous. Generalizing a definition of Beurling ([3] w167 1, 2), Herz introduced a family of spaces PLY(R") as follows (cf. [11] pp. 298--300): denote b y ~5 the set of functions ~ i n LI(0, ~ ) , ~ ~ 0 satisfying

(i)

f (t)dt

PLq(R ~) = U (L~(R~): ~ = (~ o eo)~}, y ~ 0 ,

---- O = o >_ 0 .

These spaces are Banach spaces under the respective norms

i n f IIf( o r -< 0 ,

The Beurling spaces of great interest in harmonic analysis are the spaces A P = P L 1, B P = P L ~176 ([3] p. 10). F u r t h e r spaces Kvq(Rn), K~,q(R n) were defined b y Herz to facilitate the discussion of the eLq spaces ([11] pp. 301, 302):

K~,~(~ ~) = (f: f ~ e ~/~(~n)},

K;~(R ~) : { f : f w ~+~'/~ e~L~(R~)}, -- ~ < ~ < co. (19) The following relations between the above spaces and the interpolation spaces

n p n

(L~(R'), I~,(R ))o.q;K, (I'F(R~), L~#o(R ))o,q;~ remove much of the m y s t e r y sur- rounding the Beurling-Herz spaces besides allowing use of the powerful abstract interpolation space theory in their study. I t is interesting to note that Iterz con- nected L ~ and ~L ~ b y complete interpolation (loc.cit., p. 299).

T ~ E O R ~ (5.1). When co(x)= lxl n and 1 ~ p < ~ , l ~ q ~_ ~ , VLq(R n) = (Le(Rn), L v rRn~ o,~ JJl/~-I[v,~;K q ~ P ,

p n

PLq(R") = (LV(Rn), L1/~o( R ))l/p-1/q,q;K, q > P , in particular, the Beurling spaces satisfy

2 4 8 JOHN E. GILBERT

AP(.R n) = (LP(.Rn), LP(J~'~))I_I/p,1;K, p ~ 1 , p n

Be(R ") = (LV(_R"), L~>(R )h/v,~;r, P # oo.

Furthermore, when K~q(R ~) is defined by (19),

K~~ ~) -- (LP(R"), L~(R"))o,~; K, 0 < 0 < 1.

Proof. Straightforward application of theorem (4.1).

Special cases of the characterizations in theorem (3.7) (ii), (iii) are worth noting:

r 1 1/p

f ,JIx, f ,JIx,

o iC~<_t o i~l~>_t

define equivalent norms on AP(Rn), 1 < p < 0% while

su, (} f lf(x)Fdx) lw

IxLn_<t

(20)

(21)

defines an equivalent norm on Be(Rn), 1 < p < 0o. Beurling in fact took (21) as the defining property of B*'(R~). Properties (20) and (21) have been proved or used b y several authors (cf. [9], [10], [12], [13], [18] for instance); the more general spaces considered by Sunouchi are nothing more t h a n (LR(Z), 2 L ~ ( Z ) )I/~- ~I=,e;K, 1 < fl < 2, and his characterizations are contained in theorem (3.7). The Banach algebra properties of ILv(Zn~, LPIZ ~ _{k k ! o)\ /Jo, q;K} (for appropriate values of 0, q) are studied in [1] and [2] making important use there of the interpolation space properties.

References

1. BENNETT, C. a n d GILBERT, J. E., Homogeneous algebras on the circle I I . Multipliers, D i t k i n Conditions. To appear in Ann. Inst. _Fourier.

2. - - ~ - - --,~-- Harmonic analysis of some function spaces on T n. I n preparation.

3. BEURLING, A., Construction a n d analysis of some convolution algebras. A n n . Inst. Fourier 14 (1964), 1--32.

4. BUTZER, P. L. a n d BERENS, H., Semi-groups of operators and approximation. Springer- Verlag, New York, 1967.

5. CALDER6N, A . , I n t e r m e d i a t e spaces and interpolation, the complex method. Studia Math.

24 (1964), 113--190.

6. GILBERT, J. E. a n d BENNETT, C., Interpolation space theory and harmonic analysis.

Lecture notes i n preparation.

7. GOULAOtrm, C., Prolongements de foncteurs d'interpolation et applications. Ann. Inst.

Fourier 18 (1968), 1--98.

I N T E R P O L A T I O N B E T W E E N W E I G H T E D L P - S P A C E S 249

8. GRISVAI~D, P., Commutativit4 de deux foncteurs d ' i n t e r p o l a t i o n et applications, d. Math.

Pures A p p l . (9) 45 (1966), 143--206.

9. HELSO~, H., F o u n d a t i o n s of the theory of Dirichlet Series. A c t s Math. 118 (1967), 61--77.

10. HENNIGER, J., F u n c t i o n s of b o u n d e d m e a n square, a n d generalized Fourier-Stieltjes Trans- forms. Canad. J. Math. 22 (1970), 1016--1034.

11. I ~ z , C. S., Lipschitz spaces a n d Bernstein's theorem on absolutely convergent Fourier Transforms. J. Math. Mech. 18 (1968), 283--324.

12. I~NUKAWA, M., Contractions of Fourier coefficients a n d Fourier Integrals. J. Analyse Math. 8 (1960), 377--406.

13. LEINDLER, L., Uber S t r u k t u r b e d i n g u n g e n fiir Fourierreihen. Math. Z. 88 (1965), 418--431.

14. FEETgE, J., A theory of interpolation of normed spaces. Notas lq/Iat. 39 (1963, 1968).

15. --)~-- On a n interpolation theorem of Foias a n d Lions, A c t s Sci. Math. (Szeged) 25 (1964), 255--261.

16. --))-- Espaces d ' i n t e r p o l a t i o n et thgor~me de Soboleff. A n n . Inst. 2'curler 16 (1966), 279--317.

17. --~)-- On interpolation of Lp-spaces with weight function. A c t s Sci. Math. (Szeged) 28 (1967), 61--69.

18. S~r~ouc~I, G., On the convolution algebra of Beurling. Tdhoku Math. J . 19 (1967), 303-- 310.

Received February 29, 1972. J o h n E. Gilbert

U n i v e r s i t y of Texas D e p a r t m e n t of Mathematics Austin, Texas 78712 U.S.A.